## Mixed boundary condition and uniqueness of solution

Let is uniformly elliptic in a bounded domain , , is bounded and in

(1) Let ( non-empty) and assume an interior sphere condition at each point of . Suppose satisfies the mixed boundary condition

on , on

where the vector has a non-zero normal component(to the interior sphere) at each point , then .

(2) Let satisfy an interior sphere condition, and assume that satisfies the regular oblique derivative boundary condition

on

where , is the outer normal vector. Then .

(1) Suppose constant. If there exists such that , then such that . WLOG, assume is the origin and is pointing to the negative axis. And is the interior ball at . Since is not constant, we know that . This means

Since , then . So for . So means that , this contradicts to the fact that has non-zero component on .

So in . Similarly, . So .

(2) Use the same technique.

Gilbarg, Trudinger’ book. Chapter 3, exercise 3.1.

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